Primitive root
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
An order that proves a prime
Fermat's little theorem is a test that primes pass and composites mostly fail, and it can be fooled. Run backwards, it cannot. If some number a has order exactly n − 1 modulo n, then n is prime — because only a prime has n − 1 numbers to cycle through. Checking that takes the prime factors of n − 1, which need proofs of their own, and the proofs nest into a tree that anyone can check: Pratt's certificate, which shows every prime has a short proof of being one.
The sums that obey a smaller equation
The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.
Named alongside it
The objects these essays reach for when they reach for this one.
Carmichael numberConstructible numberCosetCyclotomic polynomialFermats little theoremField extensionGalois groupOrder of an elementPrimality testPrimeRoots of unitySubgroup